It may have taken 144 years for a woman to ascend to the top of the TUC, but Frances O'Grady has every intention of using her background to help modernise and expand a trade union movement that faces one of its sternest tests of its mettle yet in an era of austerity.

In an interview with the Guardian, the 52-year-old mother of two who was elected unopposed as the TUC general secretary on Tuesday said she hoped her appointment will inspire a gender shift in the upper echelons of business and politics, as well as "nailing the stereotype" of male-dominated unions.

O'Grady will step up from deputy general secretary of the TUC – the umbrella body that represents 54 trade unions, with a collective membership of more than 6 million workers – when the current general secretary, Brendan Barber, retires at the end of the year. Widely respected within the union world as an effective negotiator who became a trade unionist while still at school, O'Grady wants the movement to make further strides towards "looking like the people it represents".

O'Grady, who raised her two children as a single parent, said: "I do know what it's like to be running around early in the morning, trying to get one kid to nursery, another kid to school and then getting to work.

"I know how difficult that is and I won't forget it. And I know what it's like now as a middle-aged woman to have adult children and also to have an elderly mother – I know what it's like to be that generation of women too. So I will bring some of that perspective."

O'Grady acknowledged that unions face a recruitment challenge as official figures published this week show trade union membership in the UK fell for the third successive year in 2011-12. There are now around 6.5 million paying union members in the UK, half the number at the movement's peak in 1979. But the former shop steward insisted that unions are more relevant than ever despite the decline, and stressed that boosting membership will be a central aim of her tenure – although she admitted that a prolonged economic downturn already made stabilising the decline a "worthy aim".

O'Grady said the success of last year's TUC-backed protest march, attended by 500,000 people in London, and a successor planned for 20 October, underlined the size of the potential recruiting ground.

"Unions have never been more relevant and we enjoy popular support for our anti-austerity campaign," she said. "You saw that at the demonstration last year. All the polls and surveys show that there are millions of people out there who would like to join a union and have not been asked. We need to convert that public sympathy into membership and organisation."

She backed a campaign by the UK's largest union, Unite, to increase its membership to 100% in workplaces where it already has a foothold and urged unions to target sectors and demographics that are under-represented, including 18 to 25 year-olds and leisure industries such as catering and hotels. "We have to ask are we relevant to young people, how can we inspire them and give them hope," said O'Grady.Unison, the largest public sector union, has signalled that public sector pay will be the next industrial battleground with the government in the wake of the pensions dispute.

Its general secretary, Dave Prentis, has warned of strike action and pledged to co-opt other unions into the fight. In an interview with the Guardian last month, Prentis said employees were suffering after two years of zero growth in public sector pay, to be followed by a sub-inflation increase of 1% in 2013-14 and 2014-15. O'Grady said the TUC would support co-ordinated strikes on pay if members vote for industrial action.

"The decision will be made by members but if unions ask us to play a role in co-ordinated action then that is what we will do. The TUC is here to support ordinary working people. We are a democracy and the job of the [TUC] general secretary is to support the decisions that working people take through a democratic ballot."

Warning of growing anger over the pincer effect of pay freezes and rising inflation, she added: "Workers in both the public and the private sectors for the last two years have had cuts in their real pay. People have less money in their pockets but the price of food, transport and child care is rising, and the more modestly paid you are, the harder inflation hits. I don't think you have to be a rocket scientist to work out that there is growing anger about pay in the workforce more generally. "

O'Grady said the pressures now applied to parents of adult children who are being forced to live at home due to lack of money. "Increasingly they're still living at home, they've got tuition fees to pay back, they get too often not real jobs but unpaid internships so the family as a whole is under pressure increasingly, and I think unless we get a shift in the government's economic policy then at some point something is going to crack."

In a critique of the government's economic policy, O'Grady said that Barack Obama's stimulus programme had proven that cuts are not the answer, while François Hollande's victory in the French presidential election confirmed a political backlash against austerity on the continent.

"The government has bet the house on cuts in public services and jobs to reduce the deficit, and it's plain for all to see that it hasn't worked. Unemployment is high, so the tax take is down and the benefits bill is up, and by its own admission the government is going to have to borrow another £150bn more over the life of the parliament than it expected. So we're now maxing out the credit card to pay the price of the government's austerity programmes."

O'Grady backed Ed Miliband following a year of friction between the Labour party leader and union bosses. Unite's Len McCluskey warned in January that Miliband's support for public sector pay restraint threatened the "destruction" of the party.

O'Grady said: "I think Ed Miliband gets the anger that ordinary people feel. I think increasingly we are seeing him and his team move to prescriptions that recognise that having decent jobs and decent living standards has to be at the heart of a new economic settlement where we don't just have an economy that serves the banks, but the banks start serving the real economy."

However, she said Labour was guilty of a "fundamental misunderstanding" of the pay issue. "Trade unions have always done deals such as a pause in pay growth, but always in return for real guarantees about protecting jobs. The problem with appearing to support the call for pay restraint is that it didn't recognise that this was being imposed. There was no agreement. There was no trade between pay and jobs.

Frances O'Grady, the TUC's first female general secretary, talks to Guardian industrial editor Dan Milmo about motherhood, why trade unions are still relevant in the 21st century and what the coalition government could learn from Obama's fiscal stimulus package. O'Grady will take over the position at the end of 2012